solve the inequality and write the answer using interval notation
50/(x+4)+1<4x
If x+4 > 0, we can multiply and not change the direction of the inequality:
4x^2 + 15x -54 > 0
(4x-9)(x+6) > 0
for x > 9/4 that is true
If x+4 < 0, e change the direction of the inequality, and have
(4x-9)(x+6) < 0
Unfortunately, whe have no values for x that satisfy the condition and the inequality.
So, x in (9/4,oo) is the only interval
To solve the inequality (50/(x+4)) + 1 < 4x, we'll go through a step-by-step process:
Step 1: Simplify the equation by first subtracting 1 from both sides:
(50/(x+4)) < 4x - 1
Step 2: Now, we want to eliminate the fraction. To do this, we can multiply both sides of the inequality by (x+4) (assuming x+4 is positive, as dividing by a negative number reverses the inequality direction):
50 < (4x - 1)(x + 4)
Step 3: Simplify the right side by distributing the terms:
50 < 4x^2 + 15x - 4
Step 4: Move all terms to one side of the inequality to form a quadratic equation:
0 < 4x^2 + 15x - 54
Step 5: We can solve this quadratic inequality in two steps by factoring and finding its roots. So, we factor the left side:
0 < (2x - 3)(2x + 18)
Step 6: Set each factor equal to zero:
2x - 3 = 0
2x + 18 = 0
Step 7: Solve each equation separately:
For 2x - 3 = 0, adding 3 to both sides gives: 2x = 3, and then dividing by 2: x = 1.5
For 2x + 18 = 0, subtracting 18 from both sides gives: 2x = -18, and then dividing by 2: x = -9
Step 8: Now, we have the critical points where the inequality can change. We can double-check these values by using a number line.
Step 9: We choose a test point in each of the intervals separated by the critical points and evaluate whether the inequality holds true or false. We can use x = 0 (between -9 and 1.5) as our test point.
Substituting x = 0 gives:
50/(0 + 4) + 1 < 4(0)
50/4 + 1 < 0
12.5 + 1 < 0
13.5 < 0
Since 13.5 is not less than 0, the inequality is false for the interval (-9, 1.5).
Step 10: Finally, we express the solution using interval notation. From our analysis, we have two intervals where the inequality holds true:
(-∞, -9) U (1.5, ∞)
Therefore, the solution to the inequality (50/(x+4)) + 1 < 4x, expressed in interval notation, is (-∞, -9) U (1.5, ∞).